Some examples of simply connected dual hyperovals

The concept of dimensional dual hyperovals was introduced by Huybrechts and Pasini [4] in 1999. Let d⩾3. It is conjectured in Yoshiara (2004) [13] that, if d-dimensional dual hyperoval S generates V(n,2), n-dimensional vector space over GF(2), then 2d−1⩽n⩽d(d+1)/2. Simply connected d-dimensional dual hyperovals are known only for n=2d−1, n=2d and n=d(d+1)/2. In this note, we will present simply connected d-dimensional dual hyperovals for n=3d−3 with d⩾4, n=4d−6 with d⩾5, and n=3d−2 with 4⩽d⩽14 satisfying some conditions.